VII. Mathematics in the Third Millennium?

Synopsis

Complicated contemporary physics. No simple equations, no hydrogen atom. Now it's many-body statistical physics. Even fundamental physical theory is like that: the quantum field vacuum is a hot bed of activity. Joke from book on many-body problems on the progress of physics, i.e., how many bodies does it take to have a problem?

The Beauty of Mathematics

I recall that at school I was good at subjects that required reasoning, not memorization. I was very good at math and physics; everything could be deduced from the basic principles. I was bad at French; there were not enough simple powerful unifying principles.

Take a good look at biology! It's a complicated mess. Are there laws of biology in the same sense as there are laws of physics? Nature is a cobbler, nature patches and reworks biological organisms, they're a mess, but they work, they survive! That's natural selection for you!

I liked physics as well as math. Look at the Bohr model for the hydrogen atom, or at the Schrödinger equation. A few simple equations explained it all!

Well, a funny thing has been happening. Math has been getting more complicated. Look at the immense computer proof-by-cases for the four-color theorem. [That's the assertion that with four colors you can paint any map on the plane in such a way that adjacent countries have different colors.] Look at the human generated but still monstrous classification of all simple groups: ten-thousand pages of proofs written by many, many mathematicians! [Roughly speaking, simple groups play the same role in group theory that the primes play in number theory. For understandable explanations of the proof of the four-color theorem and of the classification of the simple groups, see L.A. Steen, Mathematics Today—Twelve Informal Essays.]

And look at contemporary physics. Now you don't do theoretical physics writing down a simple equation and solving it analytically in closed form, like you did when I was a child. Now it's complicated computer models that you simulate on the computer to see how they behave... [See for example G.W. Flake, The Computational Beauty of Nature—Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation.]

Complicated contemporary physics! No simple equations, no hydrogen atom. Now it's many-body statistical physics. Even fundamental physical theory is like that. Even the quantum field vacuum is a hotbed of activity. Here's a joke, from a book on many-body problems, on the progress of physics, as measured by how many bodies it takes to have a problem.

Richard Mattuck, in his book he modestly refers to as Feynman Diagrams for Idiots (the official title is A Guide to Feynman Diagrams in the Many-Body Problem), sums up the progress of physics like this. How many bodies does it take to have a problem? In Newtonian physics, it was three bodies. Two gravitating point masses you can solve exactly in closed form, three, no. In general relativity two bodies get you in trouble. For a single mass point, you have the neat Schwarzchild solution, also known as the black hole. But for two bodies, it's complicated numerical work on the computer... Now in quantum field theory, even zero bodies is too much! Because the quantum mechanical vacuum is very complicated, it's a seething sea of creation and annihilation of virtual particles... You can do perturbation expansions to make estimates, but exact closed-form solutions? Forget it! [For an understandable explanation of quantum field theory, see R.P. Feynman, QED—The Strange Theory of Light and Matter... Let me give a particularly dramatic—but more technical—example. Following what's called the lattice gauge theory approach, my colleague Don Weingarten built a massively parallel super-computer just in order to do Monte Carlo estimates (estimates via statistical sampling) of Feynman path integrals (sums over all histories) in QCD (quantum chromodynamics, the theory of quarks and gluons). Each computation took about a year!]

So what will the mathematics of the future be like? Will there be wonderful new simple powerful ideas, or will things be messy and complicated as in biology? In that case, new kinds of scientific personalities will be needed to do this new kind of mathematics...

Well, it's a funny thing, but if you look back at the work of Gödel, Turing, and my own that I've presented here, this was already happening... You can already clearly see the beginning of a new kind of mathematics, a very different kind of mathematics, one that is more complicated, one that is in a way more like biology...

Here's why I say this...

A new complicated mathematics?

Peano arithmetic + first order logic as a formal axiomatic system, the code for Turing's universal machine, the interpreter for my LISP... These are very strange kinds of mathematical objects, completely different from traditional mathematical objects... Look at the primes, at the Riemann ζ function ζ(s), they're so simple... Look at a workable formal axiomatic system, at Turing's universal machine, at a LISP interpreter, they're so complicated...

So in a way, in all three cases, Gödel, Turing, and I, we already have a new ``biological'' complicated mathematics, the mathematics of the third millennium, or at least of the 21st century. [As a child I used to dream that I was in the far future, in a library, desperate to see how it had all turned out, desperate to see what science had achieved. And I would take a volume off the shelf and open it, and all I could see were words, words, words, words that made no sense at all... Writing this book brings back long-forgotten thoughts and the unusual lucidity I experience when my research is going well and everything seems inevitable.]

Integrative themes: Information, Complexity, Randomness

Look at DNA, it's biological information... Look at the new field of quantum computing and quantum information theory... Look at the title of my colleague Rolf Landauer's 1991 paper in Physics Today: ``Information is physical''...

Look at how complicated computer hardware and especially software is becoming... At the megabytes and megabytes of code one is now accustomed to have, and that you need to have, to use a computer...

Look at the human genome project, it's so much information, a huge data base of it, many huge databases... And one needs new software technology to organize it, to search it, to use it... [See D.S. Robertson, The New Renaissance—Computers and the Next Level of Civilization, for a deep information-theoretic analysis of the four different levels of civilization associated with speech, reading and writing, the printing press, and the PC, the Internet and the Web. According to Robertson, the key feature of each of these steps forward has been a substantial increase in the amount of information that can be stored, remembered and processed by the human race. And each of these jumps in information-processing power is associated with major social change.]

Look at artificial intelligence. I think it's happening, I think we're half-way there, we just don't realize it. People used to think, AI pioneers used to think, that they just needed a handful of great ideas, Nobel-prize-winning level ideas, and they would understand how human intelligence works and how to create an artificial intelligence. Instead we're getting chess playing, speech recognition and synthesis, etc., by accretion, by summing the work on hardware and software by an entire planet of hardware and software engineers... It's not a few fundamental new ideas, it's megabytes and megabytes of complicated software, that is gradually developing and evolving...

Look at some recent speculations on the nature of consciousness [D.J. Chalmers, The Conscious Mind—In Search of a Fundamental Theory, G.R. Mulhauser, Mind Out of Matter—Topics in the Physical Foundations of Consciousness and Cognition, T. Nørretranders, The User Illusion—Cutting Consciousness Down to Size] where information theory is discussed. Consciousness does not seem to be material, and information is certainly immaterial, so perhaps consciousness, and perhaps even the soul, is sculpted in information, not matter. As science fiction writers are fond of pointing out, ``soul'' is to ``body'' as ``program'' is to ``computer.''

The conventional view is that matter is primary, and that information, if it exists, emerges from matter. But what if information is primary, and matter is the secondary phenomenon! After all, the same information can have many different material representations in biology, in physics, and in psychology: DNA, RNA; DVD's, videotapes; long-term memory, short-term memory, nerve impulses, hormones. The material representation is irrelevant, what counts is the information itself. The same software can run on many machines.

Information is a really revolutionary new kind of concept, and recognition of this fact is one of the milestones of this age.

That really sums up what I have to say, what I see as the moral of the story... What I see as the broad picture... But I can't resist a few more detailed final remarks... Some final words...